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Mirrors > Home > ILE Home > Th. List > fidcenumlemrks | Unicode version |
Description: Lemma for fidcenum 6844. Induction step for fidcenumlemrk 6842. (Contributed by Jim Kingdon, 20-Oct-2022.) |
Ref | Expression |
---|---|
fidcenumlemr.dc | DECID |
fidcenumlemr.f | |
fidcenumlemrks.j | |
fidcenumlemrks.jn | |
fidcenumlemrks.h | |
fidcenumlemrks.x |
Ref | Expression |
---|---|
fidcenumlemrks |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . . . . 5 | |
2 | elun1 3243 | . . . . 5 | |
3 | 1, 2 | syl 14 | . . . 4 |
4 | df-suc 4293 | . . . . . . 7 | |
5 | 4 | imaeq2i 4879 | . . . . . 6 |
6 | imaundi 4951 | . . . . . 6 | |
7 | 5, 6 | eqtri 2160 | . . . . 5 |
8 | 7 | eleq2i 2206 | . . . 4 |
9 | 3, 8 | sylibr 133 | . . 3 |
10 | 9 | orcd 722 | . 2 |
11 | simpr 109 | . . . . . . 7 | |
12 | fidcenumlemrks.x | . . . . . . . . . 10 | |
13 | elsng 3542 | . . . . . . . . . 10 | |
14 | 12, 13 | syl 14 | . . . . . . . . 9 |
15 | fidcenumlemr.f | . . . . . . . . . . . 12 | |
16 | fofn 5347 | . . . . . . . . . . . 12 | |
17 | 15, 16 | syl 14 | . . . . . . . . . . 11 |
18 | fidcenumlemrks.jn | . . . . . . . . . . . 12 | |
19 | fidcenumlemrks.j | . . . . . . . . . . . . 13 | |
20 | sucidg 4338 | . . . . . . . . . . . . 13 | |
21 | 19, 20 | syl 14 | . . . . . . . . . . . 12 |
22 | 18, 21 | sseldd 3098 | . . . . . . . . . . 11 |
23 | fnsnfv 5480 | . . . . . . . . . . 11 | |
24 | 17, 22, 23 | syl2anc 408 | . . . . . . . . . 10 |
25 | 24 | eleq2d 2209 | . . . . . . . . 9 |
26 | 14, 25 | bitr3d 189 | . . . . . . . 8 |
27 | 26 | ad2antrr 479 | . . . . . . 7 |
28 | 11, 27 | mpbid 146 | . . . . . 6 |
29 | elun2 3244 | . . . . . 6 | |
30 | 28, 29 | syl 14 | . . . . 5 |
31 | 30, 8 | sylibr 133 | . . . 4 |
32 | 31 | orcd 722 | . . 3 |
33 | simplr 519 | . . . . . . 7 | |
34 | simpr 109 | . . . . . . . 8 | |
35 | 26 | ad2antrr 479 | . . . . . . . 8 |
36 | 34, 35 | mtbid 661 | . . . . . . 7 |
37 | ioran 741 | . . . . . . 7 | |
38 | 33, 36, 37 | sylanbrc 413 | . . . . . 6 |
39 | elun 3217 | . . . . . 6 | |
40 | 38, 39 | sylnibr 666 | . . . . 5 |
41 | 40, 8 | sylnibr 666 | . . . 4 |
42 | 41 | olcd 723 | . . 3 |
43 | fof 5345 | . . . . . . . 8 | |
44 | 15, 43 | syl 14 | . . . . . . 7 |
45 | 44, 22 | ffvelrnd 5556 | . . . . . 6 |
46 | fidcenumlemr.dc | . . . . . 6 DECID | |
47 | eqeq1 2146 | . . . . . . . 8 | |
48 | 47 | dcbid 823 | . . . . . . 7 DECID DECID |
49 | eqeq2 2149 | . . . . . . . 8 | |
50 | 49 | dcbid 823 | . . . . . . 7 DECID DECID |
51 | 48, 50 | rspc2va 2803 | . . . . . 6 DECID DECID |
52 | 12, 45, 46, 51 | syl21anc 1215 | . . . . 5 DECID |
53 | exmiddc 821 | . . . . 5 DECID | |
54 | 52, 53 | syl 14 | . . . 4 |
55 | 54 | adantr 274 | . . 3 |
56 | 32, 42, 55 | mpjaodan 787 | . 2 |
57 | fidcenumlemrks.h | . 2 | |
58 | 10, 56, 57 | mpjaodan 787 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 DECID wdc 819 wceq 1331 wcel 1480 wral 2416 cun 3069 wss 3071 csn 3527 csuc 4287 com 4504 cima 4542 wfn 5118 wf 5119 wfo 5121 cfv 5123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-suc 4293 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fo 5129 df-fv 5131 |
This theorem is referenced by: fidcenumlemrk 6842 |
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